ON THE ENDOMORPHISM SEMIGROUP
OF AN ORDERED SET
by T. S. BLYTHf
(Received 22 October, 1993)
M. E. Adams and Matthew Gould [1] have obtained a remarkable classification of
those ordered sets P for which the monoid End P of endomorphisms (i.e. isotone maps) is
regular, in the sense that for every / e End P there exists g e End P such that fgf = f.
They show that the class of such ordered sets consists precisely of
(a) all antichains;
(b) all quasi-complete chains;
(c) all complete bipartite ordered sets (i.e. given non-zero cardinals a, /3 an ordered
set Ka p of height 1 having a minimal elements and /3 maximal elements, every minimal
element being less than every maximal element);
(d) for a non-zero cardinal a the lattice Ma consisting of a smallest element 0, a
biggest element 1, and a atoms;
(e) for non-zero cardinals a,/3 the ordered set NQj3 of height 1 having a minimal
elements and )3 maximal elements in which there is a unique minimal element a0 below
all maximal elements and a unique maximal element /30 above all minimal elements (and
no further ordering);
(f) the six-element crown C6 with Hasse diagram
A similar characterisation, which coincides with the above for sets of height at most 2
but differs for chains, was obtained by A. Ya. Aizenshtat [2].
Now for every ordered set P the monoid End P can be ordered by defining
This order is compatible with the multiplication (composition) in End P. Our purpose
here is to determine precisely those ordered sets P for which the ordered semigroup
End P is regular and principally ordered in the sense that for every / e End P there exists
/ * = max{ge End P;
fgf^f}.
For the general properties of principally ordered regular semigroups we refer the reader
to [5, 6]. The main property that we shall require here is that in such a semigroup we have
ff f = / i s o t n a t / * is the biggest pre-inverse of/.
THEOREM 1. The ordered semigroup End P is regular and principally ordered if and
only if P is a dually well-ordered chain.
Proof. =$>: Suppose that End P is regular and principally ordered. Then, by [1], P
tNATO CRG 910765 is gratefully acknowledged.
Glasgow Math. J. 37 (1995) 173-178.
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T. S. BLYTH
174
must be one of the six types mentioned above. Suppose, by way of obtaining a
contradiction, that P is not a chain.
If P is an antichain or Ma or Ka^ then it is readily seen that, for a, b B P with a\\b,
the mapping fab :P-+ P defined by
\b
if JC = «;
otherwise,
belongs to End P and is idempotent. For every g e End P such that fa,bgfa,h -fa,b w e have,
applying each side to b, that fa,bg{b)~b
from which it follows that necessarily
g(b) e {a, b). Now, as a simple calculation reveals, /ft-a e End P is a pre-inverse of fab, as
is (trivially) fab itself. Consequently, /ft,fl=s/*,ft and fah^f*bWe therefore have
ftj,(b)e{a,b}
with f*,b(b)^fb.a(b) = a and fih{b)^fa,h{b)
= b, which is impossible
since a\\b. Thus P cannot be an antichain or Ma or /Ca-/3.
If P is yVa-/3 then we can choose y3 3=2 (since A^i = /C a l ). In this case, let /3] be a
maximal element distinct from /3n and consider the mapping fPu0o e End Na<p. Here
End Naifi so consider idN(t/J which is also a pre-inverse of fMo. From up0%,p0
and
which is
we obtain
and f%,,pa(Pi)
impossible since /30> /3i are maximal. Thus P cannot be NOi/3.
Finally, suppose that P is the crown
OC\
tt()
Define f:P^>P
OLi
by
f/30
/(x) = j a 2
I*
ifjc = j8i;
ifjf = oj;
otherwise.
Then / s E n d P and is idempotent. A similar argument to the above gives /*(/3))>
/(/3i) = j30 and /*(j3i) 5= i d ^ , ) = j3i, which is impossible. Thus P cannot be C6.
It follows from these observations that P must be a chain. In this case, for every
y e P define
\y
Vx
otherwise.
Then ipy e End P. Also, for y, z e P with y =£ z, define -dyz:P^>P by
otherwise.
Then -fl^ e End P.
Observe now that if x 5= y then
lpydytZ<fly(x) =
= <lty(z) =
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THE ENDOMORPHISM SEMIGROUP OF AN ORDERED SET
175
whereas if x < y then
Consequently, ifjy-dyzi{)y = ipy. It follows that, for all z^y,
therefore
we have ^
=£ i/** and
Thus we see that P has a biggest element, namely ip*(y) for every y e P.
Now let C be an ascending chain in P. By way of obtaining a contradiction, suppose
that C is infinite. Since P has a biggest element, the set C of upper bounds of C in P is not
empty. For every d E C let
Define cpd:P-+P by
otherwise.
Then <pd E End P.
Suppose now that z E Pd. Observe that if x E Pd then
whereas if x $ Pd then
<Pd$d,z<Pd(x) = <Pd$dAx) = <Pd(x) = A:.
Consequently, <pddd,z<Pd = <Pd- It follows that, for every z e Pd, we have •da.z^fS and
therefore
Thus <£?(d) is an upper bound of C and so <p%{d) e C. On the other hand,
which shows that we must have <p|(d) e ^/- From this contradiction we conclude that C
cannot be infinite. Hence all ascending chains in P are finite, so every non-empty subset of
P has a biggest element, so P is dually well-ordered.
^=: Suppose, conversely, that P is a dually well-ordered chain. Since P is quasicomplete in the sense of [1] it follows that End P is regular. In what follows we shall use
the notation
xl = {y eP;)>=£*},
xT = {y e P;y
Consider / e End P. For every x e P such that x^ D I m / ^ 0 define
Let Gybe the equivalence relation given by (x,y) e 0^if and only if f(x) =f(y). Then for
every x e P we have
/ 7 ( x ) = max{y E P;f(y) €/(*)} = maxjy e P;
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176
T. S. BLYTH
and therefore
ff*f(x)=f(x).
(VXEP)
As shown in [1, p. 197], for every x e P either x^ D Im/has a biggest element x, or
x^ fl Im/has a smallest element xF. Moreover, if for every x e Im/we choose and fix an
element h(x) ef~l{x} then h :Im/—*P so defined is an isotone injection and the mapping
g:P-*P given by
x,)
if x, exists;
otherwise,
[xF)
is an isotone pre-inverse of / For our purpose here, we observe that the same is true of
the mapping g'-.P^P given by
[xF)
[x,)
if xf exists;
otherwise.
Now since P is dually well-ordered we can choose h(x) =/ + (x) for every x e I m / and
thereby obtain from g' the mapping f*:P—>P given by
, ,
[f+(xF)
if xF exists;
v (*/)
otherwise.
f*(x) = \
Then / * E End P and is a pre-inverse of / Our objective is to show that if k e End P is
such that fkf ^ / t h e n k « / * .
For this purpose, observe that if X/ = max(x^nim/) exists then f(y)^x
implies
f(y) =£ Xi so that we have
Also, for every x e P, we have
There are two cases to consider.
(a) xF exists.
In this case we have xF=f(z) for some z B P and so
k(x) * k(xF) =fc/(z)^f+f(z)=r(xF)
=/*(*).
(b) Jtf doe5 not exist.
Here there are two sub-cases:
(bj) x^ D I m / = 0 . In this case, denoting by 1 the biggest element of P, we have
/ ( I ) =£ x in which case f+(x) = 1 and consequently
(b2) x T n i m / # 0 . In this case x^ini{f(y);f(y)^x},
the latter existing since P is
join-complete and therefore the addition of a smallest element produces a complete
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THE ENDOMORPHISM SEMIGROUP OF AN ORDERED SET
chain. Observe that the existence of x, e l m / with X/^inf{f(y);f(y)^x}
existence of f+(ini{f(y);f(y)^x}).
177
implies the
Since
it follows that
We thus have
Now iff(z)**inf{f(y);f(y)>x}
then we cannot have/(z)>x, for this would imply that
xF exists. Hence f(z)^x and consequently
It therefore follows that k(x)^f*(x).
Thus we see that in all cases k(x) =£/*(*) and therefore k =£/*. Consequently, End P
is principally ordered. •
In a principally ordered regular semigroup 5 every element has a biggest inverse [5],
that of x e 5 being x° = x*xx*. Since xx°x = x we have, in general, x° s£ JC*. We say that 5
is compact if x° = ;t* for every x e 5. For example, if 5 is completely simple then 5 is
compact; in fact, by [7, Theorem IV.2.4], if 5 is completely simple then x* e V(x) so that
x* =£ x°, whence x° = x*.
THEOREM 2. / / P is a dually well-ordered chain then the principally ordered regular
semigroup End P is compact.
Proof. Since (whenever they exist) Xj,xF e Im/, and since ff+f{x) =f(x) for every
x e P, it follows from the definition of/* that
if xF exists;
otherwise.
#*«={;Vxj
Consequently, we have
(xF)
(xj)
if xF exists;
otherwise,
and therefore f° = f* for every / e End P. D
Suppose now that P is a dually well-ordered chain. Then since End P has a biggest
element, namely the constant map n: P -* P given by n(x) = 1 for every x e P, the regular
semigroup End P is trivially strong Dubreil-Jacotin [3]. We close by using End P to settle
a question concerning perfect elements in such semigroups. Specifically, it is shown in [4]
that if S is an ordered regular semigroup that is strong Dubreil-Jacotin then the subset
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178
T. S. BLYTH
P(S) of perfect elements (i.e. those x e S such that x = x(£:x)x where £ is the bimaximum
element of 5) is a regular subsemigroup which is orthodox under certain conditions. We
show as follows, using End P where P is a dually well-ordered chain, that it is possible for
P(S) to be orthodox while S is not.
The perfect elements of End P are those isotone maps f:P—>P such that / = frtf, i.e.
those isotone maps / for which f(x) = / ( l ) for every x s P, i.e. the constant maps. It
follows that the perfect elements of End P form a left zero semigroup which is therefore
orthodox. But End P itself is not orthodox. To see this, note that since P is dually
well-ordered every p e P has a predecessor, namely
Pi = max{;t e P\ x <p}.
Define recursively p 2 = O O i , . . . ,pk = (pk-x)u-
Then each fpk
• • , and let fPik: P - » P b e given by
ifxszp;
otherwise.
= p
pk
e End P and is idempotent.
Now choose p,q e P and k ^ 2 such that qk<p
lqk
p
j
<q. Then, for every x e P,
otherwise,
ifx^q;
otherwise.
Since fp,xfq,k(q)=p and {fp,Jq,kf{q) = fP,\fq,k{p) = Pu we see that fpAfqJe is not idempotent. Hence End P is not orthodox.
REFERENCES
1. M. E. Adams and Matthew Gould, Posets whose monoids of order-preserving maps are
regular. Order 6 (1989), 195-201. See also Order 7 (1990), 105.
2. A. Ya. Aizenshtat, Regular semigroups of endomorphisms of ordered sets (Russian).
Leningrad Gos. Ped. Inst. Veen. Zap. 387 (1968), 3-11. English translation: Amer. Math. Soc.
Translations, Series 2,139 (1988), 29-35.
3. T. S. Blyth and M. F. Janowitz, Residuation Theory (Pergamon Press, 1972).
4. T. S. Blyth and E. Giraldes, Perfect elements in Dubreil-Jacotin regular semigroups.
Semigroup Forum 45 (1992), 55-62.
5. T. S. Blyth and G. A. Pinto, Principally ordered regular semigroups. Glasgow Math. J. 32
(1990), 349-364.
6. T. S. Blyth and G. A. Pinto, Idempotents in principally ordered regular semigroups,
Communications in Algebra. 19 (1991), 1549-1563.
7. M. Petrich, Introduction to semigroups (Merrill, 1973).
MATHEMATICAL INSTITUTE,
UNIVERSITY OF ST ANDREWS,
ST ANDREWS KY16
9SS,
SCOTLAND.
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